# Maximizing Wiener Index for Trees with Given Vertex Weight and Degree Sequences

## TL;DR

This paper investigates the maximum Wiener index for trees with specified vertex weights and degrees, proposing structural properties, bounds, heuristics, and an exact branch-and-bound algorithm for this complex optimization problem.

## Contribution

It introduces a comprehensive approach including structural characterization, bounds, heuristics, and an exact algorithm for maximizing the Wiener index under given constraints.

## Key findings

- Existence of an optimal caterpillar structure.
- Monotonic weight distribution along the backbone in optimal trees.
- A tight upper bound and effective heuristics for the Wiener index.

## Abstract

The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional "unweighed" Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that there exists an optimal caterpillar. If weights of internal vertices increase in their degrees, then an optimal caterpillar exists with weights of internal vertices on its backbone monotonously increasing from some central point to the ends of the backbone, and the same is true for pendent vertices. A tight upper bound of the Wiener index value is proposed and an efficient greedy heuristics is developed that approximates well the optimal index value. Finally, a branch and bound algorithm is built and tested for the exact solution of this NP-complete problem.